Convert hair particles to mesh, while keeping them deformed by armature
The new mesh needs to parented to the Armature. After you have pushed the 'convert' button - select your new mesh in Object Mode and shift select your armature. Ctrl P and choose Armature...

...select your bones in Pose Mode. Select your new 'hair' mesh and enter Weight Paint mode. You should see your bones are selected here too. Under weights click Assign automatic from bones. If need be, weight paint to adjust the effect.